Here's a geometric way of looking at it. Multiplication by a complex number can be considered as a rotation by its argument, with a scaling by its modulus. Since each $z_k$ is on the unit circle, multiplication by $z_k+z_{k+1}\;\;(\text{modulo}\;n;\; k=1\;$,...,$\;n)$ imparts a rotation by the average of the arguments of $z_k$ and $z_{k+1}$, while division by $z_{k+1}$ rotates backwards by its argument. The cyclic combination of these rotations cancels out to a null rotation, leaving only a scaling effect.